111111111×111111111 – Beauty of Mathematics

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If you multiply a number with only 1s by itself, you end up with a palindrome of the numbers in ascending and then descending order. People have called this an example of the “beauty of mathematics.” If your number has N digits of 1, then its square will be the numbers 1 to N in ascending and then descending order. The pattern continues indefinitely, although it “breaks” after 9 because of carry over. I explain why the pattern happens using the method of multiplying by lines.

Multiply by lines: https://www.youtube.com/watch?v=0SZw8jpfAk0

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panecasareccio says:

what if you have more tha 20 ones??

edwardfanboy says:

101^0 = 01
101^1 = 0101
101^2 = 010201
101^3 = 01030301
101^4 = 0104060401
101^5 = 010510100501
101^6 = 01061520150601

Saviero Sj says:


AwesomeCreeper321 says:

I Can Turn Any Number Into Four Say For Example The Number 20
20 = 6 = 3 = 5 = 4 Because 20 Has 6 Letters In It. Here Count The Letters In Twenty. Repeat This For Every Number It Always Will Equal 4

Lachezar Gorchev says:

I've just noticed that your method with the line intersections works not only for squaring but also for multiplying numbers consisting of any count of digit 1. For example 11 * 111 = 1221 , 11 * 1111 = 12221 , 111 * 1111 = 123321 …and so on.

siratthebox says:


The Space Saiyan “Ultra” God says:

Gosh i realised this aswell

Catherine Fournier says:

i feel nerdz bc i love to watch these videos

SBareSSomErMig says:

Here's another interesting thing. If you know about binomial coefficients, you will instantly be able to calculate 11^4. Why? Because 11^4 = (10+1)^4=14641

Simmilarly 101^8 = 10828567056280801

Which can be calculated by simply looking up in pascal's triangle

gameturbo131 says:

if you keep on going as far as the 28th term, the counting pattern actually continues!
123456790 123456790 123456790 098765432 098765432 0987654321
^ ^
all of the sections have all numbers 1-9 except for these. That is because all the sections in the first half have 1-9, and the very last section has 1-9, but all the others don't. This is because we carry over to the left and not the right.

gameturbo131 says:

Another way of looking at 111×111:
111×10= 01110
111×1= 00111
when it is added together it is 12321 (doesn't it look similar to the line thing?)

Jacob Scholte says:

Here's another way to do it:
1 x 1 =
111 x 111 =
x 00111

Markas Černiauskas says:

Crap this is intresting. Discovered something simmilar 5 years ago, but forgot that later.

Nui Sance says:

This happens in every base other than decimal too

Russell Schwartz says:

I always assumed this trick worked because 11 is one more than 10 (the base for our system) but I now see that that is just a coincidence. The reputation of the unit (1) is really what makes this work. Great video!

Holobrine says:

How about the pattern of the blue numbers in the middle with 10 ones or higher? They go 00, 0120, and is 012340 next?

GordonYoung says:

+MindYourDecisions can you please make a fancy video for the proof of the fundamental theorem of algebra?

Cas Dinnissen says:

Hello mindyourdecisions. i was existed to watch this new video, but i cant watch it. do you know why and how I can fix this? thanks in advance.

Иван Кольцов says:

It's really amazing. Btw, this lines drawn in 2d, will it work in 3d, with multiply 3 numbers?

Darren Shaw says:

I wish we we would switch to base 12. THEN you'd see a lot of fun patterns. And 'decimal' approximations for common ratios would be less necessary (there would still be dodecimal approximations when we have weird things but at least thirds would be 'whole' again in common math)

Tim Fischer says:

There's another pattern I have found. When you share
9 / 1
98 / 12
987 / 123
9876 / 1234

When you keep sharing this you get closer to 8. Can you explain this?

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